The Theory And Applications Of Geometric Modeling Based On Implicitly Defined Surfaces

Implicitly defined surfaces are one of the most important representations of geometric shapes in free form surface modeling. With the rapid increasing demands for dealing with more complicated shapes from industry and entertainment, geometric modeling by implicitly defined surfaces has played more and more important role in computer graphics and applied geometry. This thesis addresses several essential problems in geometric modeling of implicitly defined surfaces. The thesis is organized as follows.We briefly recall the history of Computer Aided Geometric Design at first. Several important problems are reviewed in geometric modeling based on implicitly defined surfaces.In chapter 2, we extend the mathematical model proposed by Jüttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. The idea of this algorithm is to reconstruct an implicit surface to fit the data points with detailed geometric features dynamically by knot insertion. Examples suggest that our method is more effective than Jüttler's method when the input data contains enough points and our method is able to produce reconstruction surfaces of higher quality.In chapter 3, in order to obtain a fast algorithm of computing the foot point of a given point to a spiral curve, we study the problem of circular spline fitting. An initial circular spline curve is generated by biarc interpolation at first. Then an evolution process based on a least-squares approximation is applied to the curve. During the evolution process, the circular spline curve converges dynamically to a stable shape which fits the data points well. Our method does not need any tangent information, and it is proved that the evolution process is equivalent to a Gauss-Newton-type method.In chapter 4, a new method is proposed for describing sharp features (i.e., edges and vertices) of implicitly defined surfaces. We consider an initial implicitly defined surface, which is represented as the zero set of a C~1 smooth scalar field with non-vanishing gradients. In order to represent sharp edges and vertices, this surface is augmented by adding new types of implicit representations, called edge descriptors and vertex descriptors. In our implementation, we use circular splines to describe these edge curves, since they support a fast and non-iterative closest point computation. After adding the edge and vertex descriptors to the initial scalar field, the zero set of the augmented function contains the sharp features. We apply the new representation to surface modeling by implicitly defined surfaces with sharp features and to object reconstruction.In the end, we address the problem of computing the medial axis of given boundary curves/surfaces. The distance field of given boundary curve/surface is approximated using PHT spline level by level. The last level of PHT mesh is used to represent the medial axis. After a final refinement process, we obtain a smooth representation of the medial axis of the given boundary. As an important application of our method, the offsets of the given boundary are obtained using PHT splines which approximate the distance field of the given boundary curve/surface.